
The restriction of \(F\) 
to each one of the \(S_\alpha \)  has locally constant cohomology sheaves,

All stalks are finite dimensional \(-vector spaces.\) 
The category of bounded constructible complexes of sheaves
on \(X\)  will be denoted by \(D_c^b(X)\) .
From now on, we will mostly be interested in a very special type of
bounded complexes of constructible sheaves:
the perverse sheaves for the middle perversity. Note that a perverse sheaf need not be a sheaf, but a complex of sheaves.
It is our intention for the reader to accept “being perverse”
as a good property which can be used without knowing its details.
We proceed by stating the necessary properties and results for the
intended usage.
[1]}
Let \(\)  be a perverse sheaf on a complex analytic variety \(X\) , and let \(d\)  be the dimension of the support of \(\) . For any point \(x \in X\) , the stalk cohomology groups satisfy
\(\mathcal {H}^i()_x = 0 \textnormal {, for all } i\notin [-d, 0].\) 
Note that since by definition any perverse sheaf is constructible,
its support is always a closed complex analytic subset and it makes
sense to speak of its (complex) dimension.
[1]}

If \(X\)  is a locally complete intersection of complex dimension \(d\) ,
then the shifted constant sheaf \(X[d]\)  is perverse.
Let \(g: \rightarrow  be a holomorphic function defined on a complexanalytic variety and set \) X=g-1(0)\(. Associated to \) g\(, there is the\emph {vanishing cycle functor} \)\(\phi _g : D^b_c()\rightarrow D^b_c(X) .\)\(\)
For any point \(x\in X\)  and any \(\in D^b_c()\) , the
stalk cohomology of \(\phi _gF\)  can be computed as follows
(see e.g. [1]}):
\(\mathcal {H}^i (\phi _g )_{x} = \mathbb {H}^{i+1}( B_{r}, B_{r}\cap g^{-1}(\lbrace \delta \rbrace ); ),\) 
where \(B_{r}\)  is a sufficiently small open ball in \(\) 
centered at \(x\)  and \(\delta \in \lbrace  0\rbrace \)  is small enough
with respect to \(r\) . The last result we need is essential, as it allows
to produce new perverse sheaves from old:
[1]}
The vanishing cycle functor shifted by one \(\phi _g [-1]\colon D^b_c()\rightarrow D^b_c(X)\)  takes perverse sheaves on \(\)  to perverse sheaves on \(X\) .
The ingredients we have introduced can be put together to give a simple proof of the cohomological connectivity of Milnor fibers:
Let \(g\)  be a suitable representative of a holomorphic map germ
\(({n+1},0) \rightarrow (0)\)  defined on some open subset \(\mathcal {X}\subset {n+1}\) . Take the perverse sheaf \({\mathcal {X}}[n+1]\) 
and apply the shifted vanishing cycle functor associated to \(g\)  to
obtain a perverse sheaf on \(X = g^{-1}(0)\) . The stalk cohomology at a point \(x\in X\) 
recovers the reduced cohomology of the Milnor fibre of the germ of \(X\) 
at \(x\)  as follows:
\(\mathcal {H}^i (\phi _g {\mathcal {X}}[n])_{x}&=& \mathbb {H}^{i+1}(B_r, B_r \cap g^{-1}(\delta ); {\mathcal {X}}[n])\\&=& H^{i+n+1}(B_r, B_r \cap g^{-1}(\delta ); {\mathcal {X}} ) \\ &=&H_\mathrm {sing}^{i+n+1}(B_r, B_r \cap g^{-1}(\delta ) ) \\ &=&\tilde{H}_\mathrm {sing}^{i+n}(B_r \cap g^{-1}(\delta ) ).\) 
Since the Milnor fibre of \(g\)  at a regular point of the function has trivial
reduced cohomology, we deduce that the support of \(\phi _g{\mathcal {X}}[n]\)  is contained in \(X\) . Letting \(d=\dim (X)\) , Proposition REF  gives a bound on the stalk
cohomologies at the origin
\(\mathcal {H}^i\left(\phi _g {\mathcal {X}}[n]\right)_0 \cong \tilde{H}^{i+n}_{\text{sing}}(B_r \cap g^{-1}(\delta ))= 0 \quad \text{ for all } i \notin [-d,0]\) 
which turns out to be the desired cohomological connectivity
result for the Milnor fiber of \(g\) .
As the reader will see, the proofs of our results follow the same
pattern: Find an appropriate perverse sheaf and apply the vanishing cycle
functor associated to the projection to the parameter space. By virtue
of Proposition REF , an estimate on the dimension of the
support of the sheaf will directly translate to a bound on the non-zero
degrees of the reduced cohomology of a nearby object.
Proof of Theorems ,  and 
Recall that, for \(p=n+1\) , finiteness and \(\) -finiteness are
the same and \(Y=\Delta (f)\) . Consequently, Theorem  and
Theorem  can be considered two instances of the same result for \(\) -finite germs
\(f\colon (n,0)\rightarrow (p,0)\)  with \(p\le n+1\) . For a \(\) -finite germ \(f\) , any unfolding \(F\) 
is \(\) -finite and, since \(p\le n+1\) , the discriminant \(\Delta (F)\)  is a
hypersurface in \({p}\times , by Proposition \ref {propDimOfDelta}. Inparticular, the shifted constant sheaf \) (F)[p]\( is perverse. Projecting on the unfolding parameter gives a family\)\(\Delta (F)\stackrel{\pi }{\longrightarrow }T,\)\(and one checks the fiber over \) T\( to be \) (f)\(. Applyingthe associated vanishing cycle functor gives a perverse sheaf\) (F)[p-1]\( on \) (f)\(.\) 
The support of \(\phi _\pi {\Delta (F)}[p-1]\)  is contained
in \(\mathrm {Inst}(f)\) .
Given a point away from the instability locus \(y \in \Delta (f) \setminus \mathrm {Inst}(f)\) ,
\(S:=f^{-1}(y)\cap \Sigma (f)\)  is a finite set because \(f\)  is \(\) -finite.
By Proposition REF ,
the germ \(f \colon (U,S) \rightarrow (V,y)\) 
of \(f\)  at \(S\)  is stable and thus the unfolding
\(F \colon (U \times T, S\times \lbrace 0\rbrace ) \rightarrow (V \times T,(y,0))\) 
is a trivial unfolding. Consequently, there exist unfoldings \(\Phi \)  and \(\Psi \) 
of the identity mappings \(_U\)  and \(_V\) , respectively, such that the following diagram commutes:
\(\begin{tikzcd}(U \times T, S \times \lbrace 0\rbrace  ) {r}{F} {d}{\Phi }&(V \times T, (y,0)){d}{\Psi }\\(U,S) \times (T,0){r}{f\times _{T}}& (V,y) \times (T,0).\end{tikzcd}\) 
The stalk at \(y\) 
of the sheaf of vanishing cycles of \(\pi \)  on \(\Delta (F)\)  is
\(\mathcal {H}^{i}(\phi _\pi {\Delta (F)}[p-1])_{y} &=&\mathbb {H}^{i+1}(\Delta (F)\cap B_r, \pi ^{-1}(\delta )\cap B_r;{\Delta (F)}[p-1])\\& \cong &\mathbb {H}^{i+1}((\Delta (f)\times \cap B_r, \Delta (f) \cap B_r;{\Delta (f)\times [p-1])\\&=& H^{i+p}((\Delta (f) \times  \cap B_r, \Delta (f) \cap B_r; \\&=& 0,}which finishes the proof.\) 
From the inclusion \((\phi _\pi {\Delta (F)}[p-1])\subseteq \mathrm {Inst}(f)\) 
and the hypothesis that \(\dim \mathrm {Inst}(f)\le d\) ,
applying Proposition REF  we obtain that
\(\mathcal {H}^i(\phi _\pi {\Delta (F)}[p-1])_0 = 0 \textnormal { for } i\notin [-d, 0].\) 
Theorems  and  follow, because for a good representative
the stalk at the origin is precisely
\(\mathcal {H}^i(\phi _\pi {\Delta (F)}[p-1])_{0}&=& \mathbb {H}^{i+1}(\Delta (F), \Delta (f_\delta ); {\Delta (F)}[p-1]) \\& = & \tilde{H}^{i+p-1}(\Delta (f_\delta )),\) 
where the last equality is due to the fact that \(\Delta (F)\)  is contractible.
The proof of Theorem  is very similar and thus will only be sketched. Before that, we discuss some subtleties of the analytic structure of \(D(f)\) . In order to avoid pathologies related to unfoldings, the source double point space \(D(f)\)  is given an analytic structure which need not be reduced. For our purposes, we do not need to know the details of this construction [5]}, but only the following two properties:

For any unfolding \(F=(f_t,t)\colon U\times T\rightarrow V\times T\) , the fiber over \(\delta \in T\)  of the family
\(D(F)\stackrel{\pi }{\longrightarrow }T\) 
is the complex space \(D(f_\delta )\) .

If \(f\colon (n,S)\rightarrow ({n+1},0)\)  is finite and generically one-to-one, then the space \(D(f)\)  is a hypersurface.

The first statement follows from [5]}. For the second statement, the proof of [5]} shows that \(D(f)\)  is a Cohen Macaulay subspace of \(n\)  of codimension one, hence a hypersurface.
First of all, observe that the statement of Theorem  is trivial unless the dimension \(d\)  of the instability locus is smaller than \(n-1\) . Consequently, we may assume \(f\colon (n,S)\rightarrow ({n+1},0)\)  to be a finite map-germ with \(\dim (\mathrm {Inst}(f))<n-1\) .
The map \(f\)  is generically one-to-one, because stable mappings are generically one-to-one, and the conditions of finiteness and \(\dim (\mathrm {Inst}(f))<n-1\)  imply that the preimage of the instability locus is nowhere dense.
Any unfolding \(F\)  of \(f\)  is also finite and generically one-to-one, hence \(D(F)\)  is a hypersurface in \({n+1}\)  and the sheaf \({D(F)}[n]\)  is perverse. The shifted vanishing cycle functor associated to the projection \(\pi \colon D(F)\rightarrow T\)  gives a perverse sheaf \(\phi _\pi {D(F)}[n-1]\)  on \(D(f)\) . The same argument used in Lemma  shows the inclusion
\(\phi _\pi {D(F)}\subseteq f^{-1}(\mathrm {Inst}(f)),\) 
where the dimension of \(f^{-1}(\mathrm {Inst}(f))\)  is equal to \(d\) , because \(f\)  is finite. Then the result follows
from the computation of the stalk
\(\mathcal {H}^i(\phi _\pi {D(F)}[n-1])_{0}\cong \tilde{H}^{n-1+i}(\Delta (f_\delta )).\) 

Multiple point spaces
For finite germs \(f \colon (n,0) \rightarrow (p,0)\)  with \(p> n+1\) , the
image \(Y\)  is no longer a hypersurface, and may even fail to be a complete intersection, as shown by the well known twisted cubic:
\((2,0) \rightarrow (4,0), \quad (s,t) \mapsto (s^3,s^2t,st^2,t^3).\) 
Since the constant sheaf on non complete
intersections is not necessarily perverse, we cannot follow the same reasoning as in the proof
of Theorems  and . Instead, we study the
cohomology of the disentanglement via the image computing
spectral sequence due to Goryunov and Mond [8]}:
\(E^{p,q}_1 = H^q_{}(D^{p+1}(f)) \Rightarrow H^{p+q}(Y).\) 
and the perversity of the sheaves
\(R \varepsilon ^k_* {D^k(f)}[kn -(k-1)p]\) 
on the image \(Y\) , discovered by Houston [9]}.
This involves the strict multiple point spaces \(^k(f)\) 
and their alternating cohomology which we will now discuss.
Unfortunately, there is no common agreement on the definition of multiple
point spaces and different notions are in circulation. What we will refer to
as the strict multiple points \(D^k(f)\)  is the definition used by
Goryunov, Mond [8]}, and
Houston [11]}, [12]}, [9]}.
As Example REF  shows, the strict multiple point spaces are
badly behaved in deformations: They do not specialize to fibers.
To remedy this fact, there is another, more subtle definition of
multiple point spaces \(D^k(f)\)  due to Gaffney [14]} which we will describe
in Section REF . These are what we will refer to as
simply the multiple point spaces of \(f\) .
Fortunately, the results about \(D^k(f)\)  we want to use can be adapted to the spaces \(D^k(f)\)  without difficulties.
The strict multiple point spaces \(D^k(f)\)
Let \(f\colon X\rightarrow Z\)  be a complex analytic map between holomorphic
manifolds. For \(k\ge 1\) , the \(k\) -th strict multiple point space of \(f\)  is defined to be
\(^k(f)= \textnormal {closure}\lbrace (x_{1},\dots ,x_{k})\in X^k\mid f(x_i)=f(x_j), x_{i}\ne x_{j}\text{ for all }i\ne j\rbrace .\) 
Note that \(^1(f) =X\) .
We recall the construction of the alternating complex due to Goryunov
and Mond [8]}.
\(^k(f)\)  are symmetric in the sense that group \(S_k\)  acts on them
by permutating the points \(x_1,\dots ,x_k\) . This action preserves
the fibers of the maps
\(\varepsilon ^k \colon ^k(f) \rightarrow Z, \quad (x_1,\dots ,x_k) \mapsto f(x_1) = \dots = f(x_k).\) 
As a consequence, there is an action of \(S_k\)  on the pushforward sheaf
\(\varepsilon ^k_*{^k(f)}\)  on \(Z\) .
For each \(\sigma \in S_k\) , we write the associated automorphism as
\(\sigma ^*\colon \varepsilon ^k_*{^k(f)}\rightarrow \varepsilon ^k_*{^k(f)}\) 
and define
the alternating operator
\(\colon \varepsilon ^k_*{^k(f)}\rightarrow \varepsilon ^k_*{^k(f)}\) 
by the formula
\(=\frac{1}{k!}\sum _{\sigma \in S_k}(\sigma )\sigma ^*.\) 
The image of \(\)  defines a subsheaf, which we denote by
\(\varepsilon ^k_* {^k(f)}\) .
It is clear that the map \(X^k \rightarrow X^{k-1}\) 
which forgets the \(j\) -th coordinate takes a \(k\) -multiple point to a
\((k-1)\) -multiple point. Hence, we have maps
\(\varepsilon ^{k,j}\colon ^k(f)\rightarrow ^{k-1}(f).\) 
These maps induce morphisms \(\varepsilon ^{k,j}_*\colon \varepsilon ^{k-1}_*{^{k-1}(f)}\rightarrow \varepsilon ^{k}_*{^{k}(f)}\) ,
and one can see that there is a well defined differential
\(\partial ^k\colon \varepsilon ^{k-1}_*{^{k-1}(f)}\rightarrow \varepsilon ^k_*{^k(f)},\) 
of the form
\(\partial ^k=\sum _{i=1}^k(-1)^i\varepsilon ^{k,i}_*.\) 
One can also check the equality \(\partial ^{k+1}\circ \partial ^k=0\) ,
so that we obtain a complex of sheaves \(\left(\varepsilon ^{\bullet }_*{^{\bullet }(f)},\partial ^\bullet \right)\) ,
the alternating complex.
([8]})

The augmented complex
\(0 \rightarrow Y \rightarrow \varepsilon ^{1}_*{^{1}(f)} \rightarrow \varepsilon ^{2}_*{^{2}(f)} \rightarrow \cdots \) 
is exact.
Goryunov and Mond have already argued that one has an isomorphism
\(H^i( Y, \varepsilon ^k_* {^k(f)}) \cong H^i_{}(^k(f))\) 
where the term on the right hand side is the alternating cohomology
\(\left\lbrace  [c] \in H^i_\mathrm {sing}(^k(f)) :\sigma ^* [c] = (\sigma )\cdot [c] \text{ for all } \sigma \in S_k\right\rbrace ,\) 
a subspace of the singular cohomology of \(^k(f)\) .
These considerations were taken to the derived category in
[9]}. Note that, since \(\varepsilon ^k\)  is finite,
the associated pushforward of sheaves is an exact functor
and thus in particular
\( \varepsilon ^k_* {^k(f)}= R\varepsilon _*^k {^k(f)}\) 
as complexes of sheaves on \(Y\)  in the derived category.
The same holds for their respective alternating subsheaves.
Let \(f\colon X\rightarrow Z\)  be a complex analytic mapping between two complex
analytic manifolds with \(n=\dim X\)  and \(p=\dim Z\) .
Then \(f\)  is called strictly dimensionally correct if for all \(k\ge 2\) 
the strict multiple point space \(D^k(f)\)  is either empty,
or has dimension \( k n-(k-1) p\) .
With these notations gathered
we may cite the key result due to Houston, cf. [9]},
slightly adapted to our setup:
Suppose \(f \colon X \rightarrow Z\)  is a strictly dimensionally correct complex analytic
map between complex manifolds of dimensions \(n = \dim X < \dim Z = p\) . Then
\( \varepsilon ^k_*{^k(f)}[kn-(k-1)p] \)  is a perverse sheaf.

The multiple point spaces \(D^k(f)\)
As we already mentioned before, the strict multiple point spaces discussed
in the previous section are not well behaved in families. This is illustrated
by the following example.
The cuspidal edge \(f \colon (x,y)\mapsto (x,y^2,y^3)\)  of Example REF 
and the similar germ \(f^{\prime } \colon (x,y)\mapsto (x,y^2,y^3,0)\)  of Example
 are both strictly dimensionally correct because
the maps are injective and therefore the multiple point spaces are empty.
However, the map \(F^{\prime } \colon (x,y,t)\mapsto (x,y^2,y^3+ty(x^2-1),0,t)\)  from the unfolding in
Example  is not strictly dimensionally
correct: Its multiple point spaces \(^1(F)\)  have dimension two while
the expected dimension is one.
Without the dummy variable, i.e. for the map \(f\)  as in
Example REF , not only \(f\) , but also the unfolding
\(F \colon (x,y,t)\mapsto (x,y^2,y^3+t\cdot y(x^2-t))\) 
is strictly dimensionally correct with expected dimension two for
the double point spaces. In this example we encounter another problem:
Observe that the strict double points of \(f\)  are empty
while those of \(F\)  are not. This shows the failure of specialization
\(^2(F)\cap \lbrace t=0\rbrace \ne ^2(f)\) 
and thereby illustrates the necessity to use Gaffney's multiple
point spaces.
The pathological behaviour exhibited in this example can be avoided by taking
a different definition of multiple points due to Gaffney [14]}. Our
exposition follows [20]}, where
the reader can find proofs and details omitted here.
Let \(f\colon (n,S)\rightarrow (p,0)\)  be a finite multi-germ. Since \(f\) 
is finite, hence \(\) -finite, it admits a stable unfolding [21]}
\(F=(f_t,t)\colon (n \times r,S\times \lbrace 0\rbrace )\rightarrow (p \times r,(0,0))\) 
where \(t\)  is the coordinate of the parameter space \((r,0)\) .
It is clear that a point in \(^k(F)\)  has \(t_i=t_j\) , for all \(1\le i,j\le k\) . Therefore \(^k(F)\)  can be embedded in \((n)^k\times r\) 
rather than in \((n \times r)^k\) . The
multiple point spaces of \(f\)  are defined as
\(D^k(f):=^k(F)\cap \lbrace t=0\rbrace .\) 
The space \(D^k(f)\)  does not depend on the chosen stable unfolding \(F\) .
Multiple point spaces of finite mappings between complex manifolds are
defined by patching the multiple point spaces of their corresponding
multi-germs.
To compare \(D^k(f)\)  to \(D^k(f)\) , we write the the \(k\) -th fat diagonal of \(X\) as
\(\Delta ^k = \lbrace  (x_1,\dots ,x_k) \in X^k \mid x_i = x_j \text{ for some } i \ne j \rbrace .\) 
(Properties of the multiple point spaces)

Let \(f\colon X\rightarrow Z\)  be a finite mapping between complex manifolds. Then

the spaces \(D^k(f)\) 
and \(^k(f)\)  satisfy the relation
\(^k(f)=\overline{D^k(f)\setminus \Delta ^k};\) 

if \(f\)  is stable, then \(D^k(f)=^k(f)\) ;

unlike \(D^k(f)\) , the spaces \(D^k(f)\)  behave well under
deformations in the sense that, for any unfolding \(F\colon X\times T\rightarrow Z\times T\) , the family \(\pi \colon D^k(F)\rightarrow T\)  satisfies
\(D^k(f_{\delta })=\pi ^{-1}(\delta ),\text{ for any }\delta \in T.\) 

Just as in the case of \(^k(f)\) , the spaces \(D^k(f)\)  are \(S_k\) -invariant spaces endowed with finite maps \(\varepsilon ^{k,j}\colon D^k(f)\rightarrow D^{k-1}(f)\) , giving rise to a complex
\(\left(\varepsilon ^{\bullet }_*{D^{\bullet }(f)},\partial ^\bullet \right).\) 
We wish to replace \(^k(f)\)  by our favourite \(D^k(f)\)  in the study
of \(Y\) . While the spaces \(D^k(f)\)  and \(^k(f)\)  and hence also
the sheaves \(\varepsilon _*^k {D^k(f)}\)  and \(\varepsilon ^k_* {^k(f)}\) 
may differ, the corresponding alternating sheaves do not:
For any finite map \(f\)  between complex manifolds, the complexes \(\left(\varepsilon ^{\bullet }_*{D^{\bullet }(f)},\partial ^\bullet \right)\)  and \(\left(\varepsilon ^{\bullet }_*{^{\bullet }(f)},\partial ^\bullet \right)\)  are equal.
Note that with this lemma we may replace \(\varepsilon ^k_* {^k(f)}\)  by
\(\varepsilon _*^k {D^k(f)}\)  in Proposition REF .
The idea behind the proof of Lemma REF  is not
new and appears already in [8]}, as well as [9]} and
[12]}. Due to the lack of a citeable reference, we
give a self contained argument.
We verify that the stalks of both sheaves coincide at an arbitrary point
\(y \in Y\) . Fix one such point and let
\(P := \left( \varepsilon ^k \right)^{-1}(y) \subset D^k(f)\) 
be its preimage under the finite map \(\varepsilon ^k \colon D^k(f) \rightarrow Y\) .
Set \(P^{\prime } := ^k(f) \cap P\)  to be the preimage in the subspace
\(^k(f)\) . According to Proposition REF ,
every point \(x \in P \setminus P^{\prime }\)  has to be contained in the fat diagonal
\(\Delta ^k\) . Now the pushforward of \({D^k(f)}\)  along \(\varepsilon ^k\) 
decomposes as
\(\varepsilon _*^k {D^k(f)} &=&\left( \bigoplus _{x \in P^{\prime }} {D^k(f),x} \right)\oplus \left( \bigoplus _{x \notin P^{\prime }} {D^k(f),x} \right).\) 
The alternating operator clearly respects this decomposition
since \(P^{\prime }\)  is itself an \(S_k\) -invariant subspace.
Note that the first summand is nothing but the stalk of
\(\varepsilon _*^k {^k(f)}\)  by definition of \(P^{\prime }\) . Thus,
we may conclude the proof by showing that the second summand
has no alternating part.
To see the latter, let \(x = (x_1,\dots ,x_k) \in P \setminus P^{\prime }\) 
be an arbitrary point and \(c \in {D^k(f),x}\)  an element of
its associated stalk. Since \(x\)  has to be a diagonal point,
there exists one pair of indices \(0< i<j \le k\)  such that
\(x_i = x_j\) . Let \(\tau \in S_k\)  be the corresponding transposition.
Then we have
\(- c = \tau * c = (\tau *c) = c,\) 
because \(c\)  is alternating and \(\tau \)  takes the point \(x\)  to
itself. Thus, \(c = 0\)  for every element \(c\)  of any stalk
of \({D^k(f)}\)  at points outside \(^k(f)\) .
Parallel to (REF ) we have the equality
\(H^i(Y,\varepsilon ^k_*{D^k(f)})=H^i_{}(D^k(f),.\) 
This shows indirectly that \(^k(f)\)  and \(D^k(f)\)  have the
same alternating cohomology, thanks to the equality \(\left(\varepsilon ^{\bullet }_*{D^{\bullet }(f)},\partial ^\bullet \right)=\left(\varepsilon ^{\bullet }_*{^{\bullet }(f)},\partial ^\bullet \right)\) 
proved in Lemma REF .
As in the case of \(^k(f)\) , if \(f\colon X\rightarrow Z\)  is a mapping between two complex
analytic manifolds with \(n=\dim X\)  and \(p=\dim Z\) , then every irreducible component of \(D^k(f)\)  has dimension at least \(kn-(k-1)p\) . The use of \(D^k(f)\)  rather than \(^k(f)\)  calls for the following adaptation:
A finite map-germ \(f\colon (n,S)\rightarrow (p,0)\)  with \(p>n\)  is
dimensionally correct if for all \(k\ge 2\)  the multiple point
space \(D^k(f)\)  is either empty or has dimension \(k n-(k-1) p\) .
It is clear that every unfolding \(F\)  of a finite map-germ \(f\)  must be finite as well. Moreover, any stable unfolding of \(F\)  is a stable unfolding of \(f\)  as well. From this observation, one concludes easily the following result:
If a map-germ \(f\)  is dimensionally correct, then every unfolding \(F\)  of \(f\)  is dimensionally correct.
The previous assertion
does not hold if one replaces the property of being dimensionally
correct by that of being strictly dimensionally correct (see Example
REF ). Being dimensionally correct implies being
strictly dimensionally correct, but the converse is not true, as shown
again by the germ \((x,y)\mapsto (x,y^2,y^3,0)\) .

The image computing spectral sequence
Let \(f\colon (n,0)\rightarrow (p,0)\)  be a finite map-germ with \(p>n\)  and let
\(F\)  be a one-parameter unfolding of \(f\) . We wish to compute the
reduced cohomology of the disentanglement \(Y_\delta \)  of a good representative
of \(F\) . Goryunov and Mond have shown [8]} how
to do this in terms of the alternating complex. We are basically going
to rephrase their argument in the derived category so that we can bring the
perversity of the alternating sheaves, Theorem REF , into the
picture.
In the derived category, Proposition REF 
and Lemma REF 
assert that the constant sheaf \({\mathcal {Y}}\)  on the image \(\mathcal {Y}\)  of the unfolding \(F\) 
is quasi-isomorphic to the alternating complex
\(\left( \varepsilon ^\bullet _* {D^k(F)}, \partial ^\bullet \right)\) 
and we may thus replace \({\mathcal {Y}}\)  by this complex.
If we let \(\pi \colon \mathcal {Y} \rightarrow T\)  be the projection onto the unfolding
parameter, then
the reduced cohomology of the disentanglement \(Y_\delta \) 
are the vanishing cycles of \(\pi \)  at the origin:
\(\tilde{H}^{i}(Y_\delta ) &\cong & \mathcal {H}^i( \phi _\pi \left( {\mathcal {Y}}\right))_0 \\&\cong & \mathcal {H}^i(\left(\phi _\pi (\varepsilon ^\bullet _* {D^\bullet (F)}) \right)_0.\) 
The nearby and the vanishing cycles of
\(\pi \colon \mathcal {Y} \rightarrow T\)  are computed
using the well known diagram
\({Y_0 @{^{(}->}[r]^i [d] &\mathcal {Y} [d]^\pi &\mathcal {Y} \setminus Y_0 @{_{(}->}[l]_j [d]&Y_\delta \times S [l]_\rho [d] \\\lbrace 0\rbrace  @{^{(}->}[r] &T &T^* @{_{(}->}[l] &S [l]^{\exp }}\) 
for a good representative \(F\)  of a 1-parameter family and its image
\(\mathcal {Y}\) . Here, \(\exp \colon S \rightarrow T^*\)  is the universal cover
of the punctured unit disc and
\(Y_\delta \times S\)  the trivialized fiber product over the infinite strip \(S\) ,
cf. [1]}.
For a good representative \(F\)  of a 1-parameter unfolding of
a \(\mathcal {K}\) -finite map
\(f \colon (n,0) \rightarrow (p,0)\)  with \(n<p\) 
there is a spectral sequence with first page
\(E^{i,j}_1 &=& \mathcal {H}^i\left(\phi _\pi \left(\varepsilon ^{j+1}_* {D^{j+1}(F)}\right)\right)_0\) 
converging to the reduced cohomology of the disentanglement \(Y_\delta \)  of \(f\)
\(\tilde{H}^{i+j}( Y_\delta ) = \mathcal {H}^{i+j}(\phi _\pi {\mathcal {Y}})_0.\) 
Here \(\pi \colon \mathcal {Y} \rightarrow T\)  is the projection of the image \(\mathcal {Y}\)  of
\(F\)  to the deformation parameter.
The standard procedure to obtain a quasi-isomorphism of the complex
\(\varepsilon ^\bullet _* {D^\bullet (F)}\) 
with a complex of injectives is to construct a double complex of injectives
\(I^{\bullet ,\bullet }\)  where each column \(I^{\bullet ,j}\)  resolves the sheaf
\(\varepsilon _*^{j+1} {D^{j+1}(F)}\) , cf. [8]}.
Then the original complex is quasi-isomorphic to the total complex \(\mathrm {Tot}(I^{\bullet ,\bullet })\) 
of this double complex \(I^{\bullet ,\bullet }\) .
The nearby cycles of \(\pi \)  are defined as
\(\psi _\pi \left( \varepsilon _*^\bullet {D^\bullet (F)} \right):=i^{-1} R(\rho \circ j)_* (\rho \circ j)^{-1}\left( \varepsilon _*^\bullet {D^\bullet (F)} \right).\) 
This is a composition of three functors. The first one,
\((\rho \circ j)^{-1}\) , takes injective sheaves on \(\mathcal {Y}\) 
to injective sheaves on the open subset \(\mathcal {Y} \setminus Y_0\) ,
see [28]} and the third one,
\(i^{-1}\) , is an exact functor. To compute the nearby cycles
of \(\varepsilon _*^\bullet {D^\bullet (F)}\) —which involves the derived pushforward in the middle—we may therefore apply the functor \(i^{-1} (\rho \circ j)_* (\rho \circ j)^{-1}\) 
to \(\mathrm {Tot}(I^{\bullet ,\bullet })\) .
Note that applying this functor
to a seperate column \(I^{\bullet ,j}\)  yields the nearby cycles of the single sheaf
\(\varepsilon _*^{j+1} {D^{j+1}(F)}\)  considered as a complex of sheaves
concentrated in a single degree.
The vanishing cycles \(\phi _{\pi }{\mathcal {Y}} \) 
of \(\pi \)  are defined as the mapping
cone of the comparison morphism \(c\)
\({i^{-1} {\mathcal {Y}} [r]^c &\psi _{\pi } {\mathcal {Y}} [r] &\phi _\pi {\mathcal {Y}} [r]^>>>>>{[+1]} &.}\) 
Following Definition REF ),
the image \(\mathcal {Y}\)  of a good representative is connected, hence the
stalk of \(i^{-1}{\mathcal {Y}}\) 
at the origin is simply the vector space \(, concentrated in degree zero. It is easy tocheck that on this single nontrivial term the comparison morphism \) c\(is always an inclusion which makes the vanishing cycles coincidewith the reduced cohomology of the disentanglement, cf. (\ref {eqn:vanishingCycleFunctorStalk}).\) We may again replace \({\mathcal {Y}}\)  by the
alternating complex \(\varepsilon _*^\bullet {D^\bullet (F)}\)  and
subsequently by \(\mathrm {Tot}(I^{\bullet ,\bullet })\)  in the distinguished triangle of
the comparison morphism:
\({i^{-1}\mathrm {Tot}(I^{\bullet ,\bullet })[r]^<<<<c &i^{-1} (\rho \circ j)_* (\rho \circ j)^{-1}\mathrm {Tot}(I^{\bullet ,\bullet }) [r] &\phi _\pi \mathrm {Tot}(I^{\bullet ,\bullet })[r]^>>>>>{[+1]} &.}\) 
Now note that the functor \(i^{-1} (\rho \circ j)_* (\rho \circ j)^{-1}\) 
commutes with taking the total complex in a natural way.
Thus we may identify the distinguished triangle of the comparison
morphism \(c\)  with the total complex of the following “comparison morphism of
double complexes”
\({i^{-1} I^{\bullet ,\bullet }[r]^<<<<{\tilde{c}} &i^{-1} (\rho \circ j)_* (\rho \circ j)^{-1}I^{\bullet ,\bullet } [r] &\text{Cone}(\tilde{c}) [r]^>>>>>{[+1]}&}\) 
which turns the vanishing cycles \(\phi _\pi \mathrm {Tot}(I^{\bullet ,\bullet })\) 
into the total complex of the double complex \(\text{Cone}(\tilde{c})\) .
The spectral sequence in question is now the spectral sequence of
this double complex \(\text{Cone}(\tilde{c})\) . A column-wise
inspection of this double complex reveals that
we indeed find the vanishing cycles of the alternating sheaves on the
first page.
From the fact that the spaces \(D^k(F)\)  are empty for \(k\)  big enough, it follows that the spectral sequence collapses at a certain page. Since we chose a field for the coefficients in
cohomology, the infinity page determines the cohomology of the total complex. To be precise, there is an isomorphism of vector spaces
\(\mathcal {H}^\ell (\phi _\pi {\mathcal {Y}})_0\cong \bigoplus _i E^{i,\ell -i}_\infty .\) 
Here we come back to the immersion bi-germ \(f\colon (n,S)\rightarrow ({n+1},0)\)  associated to a hypersurface \(X=V(g)\) , introduced in Example , where we find a particularly simple spectral sequence.
First of all, there are no triple or higher multiplicity points. This is because the stable unfolding \(F\)  of \(f\)  (or any unfolding of \(f\)  for that matter) is also a bi-germ of an immersion, hence it cannot map more than two points to one. This gives \(^k(F)=\emptyset \) , and thus \(D^k(f)=\emptyset \) , for all \(k\ge 3\)  . Consequently, the nonzero terms of the first page
\(E_1^{\bullet ,\bullet }\)  are concentrated in the second column, corresponding to the alternating cohomologies \(H^i_{}(D^2(f_\delta ))\) .
The fact that there are no triple points implies also that the maps \(D^2(F)\rightarrow D(F)\)  and \(D^2(f_\delta )\rightarrow D(f_\delta )\)  are homeomorphisms. Recall also from Example 
that \(D(f_\delta )=M_1\sqcup M_2\) , where \(M_1\)  and \(M_2\)  are copies of the Milnor fibre of \(X=V(g)\) . The homeomorphism \(D^2(f_\delta )\rightarrow D(f_\delta )\)  transforms the action of the generator \(\sigma \in S_2\)  into the map taking a point \(x\in M_1\)  to the same point in \(M_2\) , and vice-versa. In particular, after identifying \(D^2(f_\delta )\)  with \(M_1\sqcup M_2\) , we can choose a system of generators of the cohomology \(H^i(D^2(f_\delta ))\)  consisting of some cocycles \(c_i\)  generating the cohomology of \(M_1\)  and their corresponding cocycles \(\sigma \cdot c_i\) , which generate the cohomology of \(M_2\) . It now follows immediately that the alternating part is generated by the cocycles \(c_i-\sigma \cdot c_i\in H^i_{}(D^2(f_\delta ))\) . The linear map extending \(c_i\mapsto c_i-\sigma \cdot c_i\)  gives an isomorphism
\(H^i(M)\cong H^i_{}(D^2(f)).\) 
Finally, since \(E_1^{\bullet ,\bullet }\)  contains a single nonzero column, the spectral sequence collapses at page one, that is, \(E_\infty ^{\bullet ,\bullet }=E_1^{\bullet ,\bullet }\) . Now, the isomorphism \(\mathcal {H}^\ell (\phi _\pi {\mathcal {Y}})_0\cong \bigoplus _i E^{i,\ell -i}_\infty \)  takes the form
\(\tilde{H}^i(M)\cong \tilde{H}^{i+1}(Y_\delta ),\) 
as Remark REF  claimed.
The previous example has a more visual version in homology (Figure REF  depicts the case of \(g=x^2+y^2\)  which, after a change of coordinates, is the same as in Figure REF , but with a real picture better suited for the present discussion). We know that \(D(f_\delta )\)  consists of two copies \(M_1\)  and \(M_2\)  of \(M\) , contained in two copies \(U_1\)  and \(U_2\)  of an open ball \(U\in n\) . Now let \(c\)  be a cycle in \(M\)  and let \(c_i\)  be the corresponding copies in \(U_i\)  (the two circles one the left side in Figure REF ). Since \(U\)  is contractible, there are chains \(a_i\)  in \(U_i\)  with boundary \(\delta a_i=c_i\) (the green and blue disks). Observe that \(\partial a_i\)  is supported on \(M_i\) , which is the double point space of \(f_\delta \) , and that \(f_\delta \)  glues \(a_1\)  and \(a_2\)  along the boundaries \(\partial a_i\) . After changing the sign of \(a_2\)  if necessary, we observe that \(f_*(a_1+a_2)\)  is an \((i+1)\) -dimensional cycle on \(Y_\delta \)  (the blue and green cycle on the right side). The desired isomorphism is determined by \(c\mapsto f_*(a_1+a_2)\) .
<FIGURE>
Proof of Theorem 
In this section we will use the notation of Theorem  and
impose the hypothesis found there,
that is, \(f\colon (n,0) \rightarrow (p,0)\)  is a dimensionally correct
map-germ with \(p>n\)  and instability locus \(\mathrm {Inst}(f)\)  of dimension \(d\) ,
\(F\colon W \rightarrow V\times T\)  is a good representative of a one-parameter unfolding of \(f\) 
and \(\mathcal {Y}=F\)  is the image of \(F\) .
We intend to control the stalk cohomology of \(\phi _{\pi }{\mathcal {Y}}\) ,
i.e. the reduced cohomology of the disentanglement, by means of the first page
of the spectral sequence in Lemma REF .
Thus, we have to bound the vanishing cohomology of the alternating sheaves
\(\varepsilon _*^{j+1} {D^{j+1}(F)}\)  on the image and we will do so by
exploiting the perversity of their respective shifts and estimating
the dimension of the support of their vanishing cycles.
The support of \(\phi _\pi \left(\varepsilon ^k_*{D^k(F)}\right)\) 
is contained in \(\mathrm {Inst}(f)\) .
Take a point away from the instability locus
\(y\in Y\setminus \mathrm {Inst}(f)\)  and let \(S=f^{-1}(y)\) . Since \(f\) 
is stable at \(y\) , the unfolding \((F,S\times \lbrace 0\rbrace )\)  of the germ
\((f,S)\)  is trivial. Therefore, there exist an unfolding \(\Phi \)  of
\((_{n},S)\)  and an unfolding \(\Psi \)  of \((_{p},0)\) , such that
\((F,S\times \lbrace 0\rbrace )=\Psi ^{-1}\circ ((f,S)\times _{(T,0)})\circ \Phi .\) 
We choose representatives \(F, f, \Phi \)  and \(\Psi \)  satisfying
the equality
\(F=\Psi ^{-1}\circ (f\times _{T})\circ \Phi .\) 
As one may easily check, this makes the multiple
points \(D^k(F)\)  and \(D^k(f\times _{T})\)  isomorphic via
\(\Phi ^{-1}\times \stackrel{k}{\dots }\times \Phi ^{-1}\) . Moreover,
the multiple points in \(D^k(f\times _{T})\)  are of the form
\(((x_1,t),\dots ,(x_k,t))\) . Forgetting all but one of the copies of \(t\) 
gives an isomorphism [20]}
\(\Omega \colon D^k(f\times _{T})\rightarrow D^k(f)\times T.\) 
In turn, the spaces \(\)  and \(Y\times T\)  are isomorphic via \(\Psi \) .
There are two geometric considerations where the \(k\) -fold product structure
of \(\Phi ^{-1}\times \stackrel{k}{\dots }\times \Phi ^{-1}\)  plays a role:
First of all, the isomorphisms and the \(\varepsilon ^k\)  mappings are compatible
in the sense that there is a commutative diagram
\(\begin{tikzcd} \phantom{Aaa} D^k(F)\phantom{Aaa}{r}{\Omega \circ (\Phi ^{-1}\times \stackrel{k}{\dots }\times \Phi ^{-1})}{d}{\varepsilon ^k} &\phantom{Aaa}D^k(f)\times T\phantom{Aaa}{d}{\varepsilon ^k\times _T}\\\phantom{Aaa}\phantom{Aaa}{r}{\Psi }& \phantom{Aaa}Y\times T.\phantom{Aaa} \end{tikzcd}\) 
Second, the map
\(\Omega \circ (\Phi ^{-1}\times \stackrel{k}{\dots }\times \Phi ^{-1})\)  is
compatible with
the action of the symmetric group \(S_k\)  on both
\(D^k(F)\)  and on \(D^k(f)\times T\)  (the latter
induced by the action of \(S_k\)  on \(D^k(f)\) ).
This makes the construction of the \(\)  operators on
both sides equivalent and the pushforward
\(\Psi _*\)  takes the sheaf \(\varepsilon ^k_*{D^k(F)}\)  on \(\) 
to the sheaf \((\varepsilon ^k\times _T)_*{D^k(f)\times T}\) 
on \(Y\times T\) .
We can now show that \(y\)  is not in the support of \(\phi _\pi \left( \varepsilon ^k_*{D^k(F)}\right)\)  by checking the vanishing of the stalk
cohomology: Take a suitable ball \(B_r\)  in \(V\)  around
\(y\) . Then, for \(\delta \)  sufficiently small we find
\(& & \mathcal {H}^i\left(\phi _\pi \left(\varepsilon ^k_*{D^k(F)}\right) \right)_{y}\\&=& \mathbb {H}^{i+1}\left(\mathcal {Y}\cap B_r, \mathcal {Y}\cap \pi ^{-1}(\delta )\cap B_r;\varepsilon ^k_*{D^k(F)} \right)\\& \cong & \mathbb {H}^{i+1}\left((Y\times T)\cap B_r, (Y\times \lbrace \delta \rbrace )\cap B_r ; \varepsilon ^k_*{D^k(f)\times T} \right)\\&=& 0.\) 
The vanishing in the last line follows from the fact that
due to the product structure clearly
\((Y \times T) \cap B_r\)  retracts onto the fiber \((Y \times \lbrace \delta \rbrace ) \cap B_r\) 
for \(r \gg \delta > 0\) .
Now we can determine where the non-trivial entries of the first
page of the spectral sequence in Lemma REF 
are concentrated.
Let \(F\)  be a dimensionally correct one-parameter unfolding of a germ \((n,0)\rightarrow (p,0)\)  with \(p> n\) .

For every integer \(k\ge 2\) , the nonzero cohomologies
\(\mathcal {H}^i\left(\phi _\pi \left( \varepsilon ^k_*{D^k(F)}\right) \right)_{0}\) 
are concentrated in degrees \(i\ge 0\)  with
\(kn-(k-1)p-d \le i \le kn-(k-1)p.\) 
In the case \(k=1\)  the above vanishing cycles are all zero.
Since \(F\)  is dimensionally correct, the sheaves
\(\varepsilon ^k_*{D^k(F)} [kn-(k-1)p+1]\) 
are perverse, and, by virtue of
Theorem REF ,
so are the sheaves
\(\phi _\pi \left(\varepsilon ^k_*{D^k(F)}\right)[kn-(k-1)p]\) 
on \(Y_0\) . Since these sheaves
are supported on a space of dimension at most \(d\) , it follows
from Proposition REF  that
the non trivial cohomologies of their stalks at the origin are
concentrated in degrees \(i\)  with \(-d \le i \le 0\) .
Shifting everything back by \(kn-(k-1)p\) , we obtain the desired bounds
for \(k\ge 2\) .
Note that, since the alternating cohomology
\(\mathcal {H}^i\left(\phi _{\pi }\left( \varepsilon _*^k {D^k(F)} \right) \right)_0&=& \tilde{H}^i_{} (D^k(f_\delta ))\subset \tilde{H}^i(D^k(f_\delta ))\) 
is a subspace of the singular reduced cohomology of the disentanglement,
there can be no contributions in negative degrees.
For \(k=1\)  the disentanglement always provides a product structure
\((D^1(F),(0,0)) \cong (X \times T,(0,0))\) 
and therefore
\(\mathcal {H}^i\left(\phi _\pi \left(\varepsilon ^1_*{D^1(F)}\right)\right)_{0}= H^i(X\times T,X\times \delta )=0.\) 
This finishes the proof.
The proof follows by the same
spectral sequence argument as in [30]}:
The concentration of cohomology follows immediately
from the isomorphism
\(\mathcal {H}^\ell (\phi _\pi {\mathcal {Y}})_0\cong \bigoplus _i E^{i,\ell -i}_\infty \) 
from Remark REF 
and the vanishing of the entries of the first page
coming from Proposition .
Let us give an example which illustrates the general
situation:

Let \(f\colon ({16},0)\rightarrow ({21},0)\) 
be a dimensionally correct map whose instability locus has
dimension \(d=2\) . According to Proposition , the possibly nonzero entries in the first page of the spectral sequence \(E_1^{i,j}\)  from Lemma
REF  are the following:
\(\begin{array}{ccccccccccccccccccccccccccccc}\multicolumn{1}{c|}{i \uparrow } && & & \\\multicolumn{1}{c|}{} && & & \\\multicolumn{1}{c|}{11} && H^{11}_{}(D^2(f_\delta )) & & & \\\multicolumn{1}{c|}{10} &\phantom{\vdots } & H^{10}_{}(D^2(f_\delta )) & & \\\multicolumn{1}{c|}{ 9} &\phantom{\vdots } & H^{9}_{}(D^2(f_\delta ))& & \\\multicolumn{1}{c|}{ 8} &\phantom{\vdots } & & &\\\multicolumn{1}{c|}{ 7 }&\phantom{\vdots } & & &\\\multicolumn{1}{c|}{ 6} &\phantom{\vdots } & & H^{6}_{}(D^3(f_\delta )) &\\\multicolumn{1}{c|}{ 5 }&\phantom{\vdots } & & H^{5}_{}(D^3(f_\delta )) &\\\multicolumn{1}{c|}{ 4} &\phantom{\vdots } & & H^{4}_{}(D^3(f_\delta )) &\\\multicolumn{1}{c|}{ 3} &\phantom{\vdots } & & &\\\multicolumn{1}{c|}{ 2} &\phantom{\vdots } & & &\\\multicolumn{1}{c|}{ 1} &\phantom{\vdots } & & & H^{1}_{}(D^4(f_\delta ))\\\multicolumn{1}{c|}{ 0} &\phantom{H^{10}_{}(D^2(f_\delta ))} & \phantom{\vdots } & & H^{0}_{}(D^4(f_\delta )) \\\hline \multicolumn{1}{c|}{}& 0 & 1 & 2 & 3 &\underset{j}{\longrightarrow }\end{array}\) 
In this case, the sequence necessarily collapses on this first page. In
general, the differentials between the nonzero entries on the first
page can lead to cancellations for the following ones and on the limit
page. In either case, the positions of the nonzero entries on the first
page give a bound on the nonzero entries of
\(\bigoplus _i E_\infty ^{i,k-i} \cong \mathcal {H}^k(\phi _\pi {\mathcal {Y}})_0\) .
For this particular example we see that the non-trivial cohomologies
of a disentanglement \(Y_\delta \)  are concentrated on the degrees \(\ell \in \lbrace 3,4\rbrace \cup \lbrace 6,7,8\rbrace \cup \lbrace 10,11,12\rbrace .\)

Monodromy for disentanglements
Our study of the monodromy for disentanglements in
Theorem  will be based on the monodromy
on the multiple point spaces and their alternating cohomology. This
requires an adaptation of the definitions to fibrations with
group actions.
Note that any stable one parameter unfolding
\(F \colon (n,0) \times (0) \rightarrow (p,0)\times (0)\) 
of a finite map
germ \(f\)  with \(p>n\)  as in Theorem 
gives rise to analytic fibrations
\(\pi \circ \varepsilon ^k \colon D^k(F) \cap (\varepsilon ^k)^{-1}(B \times D\setminus \lbrace 0\rbrace )\rightarrow D\setminus \lbrace 0\rbrace \) 
for suitable choices of a representative \(F\) , a ball \(B \subset p\)  in
the target of \(f\) , and a disc \(D \subset  inthe parameter space.By construction, the symmetric group \) Sk\( acts fiberwise on the multiple pointspace \) Dk(F) (k)-1(BD)\(, for each \) k>0\(.\) As already noted by K. Houston in [11]},
there exist equivariant
local trivializations.
We may therefore suppose that the trivialization of the pullback of the fibration
(REF ) to the universal cover
\(\exp \colon S \rightarrow D\setminus \lbrace 0\rbrace \)  is \(\mathfrak {S}_k\) -equivariant and
that the parallel transport of the fiber
\(h \colon D^k(f_\delta ) \rightarrow D^k(f_\delta )\) 
along a closed, counterclockwise loop in \(D^*\)  around the origin
commutes with the \(\mathfrak {S}_k\) -action on \(D^k(f_\delta )\) .
It is easy to see that this implies that also the maps induced by \(h\)  on cohomology
\(h^i \colon H^i( D^k(f_\delta )) \rightarrow H^i( D^k(f_\delta ) )\) 
restrict to operators on the alternating cohomology
\(H^i_{}(D^k(f_\delta ))\) . We therefore have
a well defined monodromy action
\(h \colon \phi _\pi \left(\varepsilon _*^k {D^k(F)} \right)\rightarrow \phi _\pi \left(\varepsilon _*^k {D^k(F)} \right)\) 
on the vanishing cycles of the alternating sheaves. As we shall see now, this gives the vanishing cycles of
the alternating sheaves the structure of a \(s, s^{-1}]\) -module where
\(s\)  is a formal variable standing for the monodromy automorphism \(h\) .
To relate monodromy to our usual spectral sequence, it is useful to introduce a point of view about eigenvalues and Jordan blocks slightly different from that of the introduction: Consider the monodromy of a fibration \(E\xrightarrow{} D^*\)  over the punctured disk
with fibre \(F\) , and assume that the fibration was obtained as a small representative of an analytic germ \((E,0)\rightarrow (D,0)\) , as in [32]}. Consider the one-variable Laurent polynomial ring \(R:=s, s^{-1}].\)  The monodromy action gives each cohomology \(H^i(F)\)  an \(R\) -module structure, by declaring that multiplication by \(s\)  acts on \(H^i(F)\)  by the monodromy automorphism \(h^i\colon H^i(F)\rightarrow H^i(F)\) . It is well known that \(F\)  is a homotopy equivalent to a finite CW-complex, and that each monodromy automorphism \(h^i\)  is idempotent. As a consequence, each \(H^i(F)\)  is a finitely generated torsion \(R\) -module. Note that \(R\)  is a principal ideal domain so that according to the structure theorem, every finitely generated torsion \(R\) -module \(A\)  can be uniquely decomposed as
\( A\cong \oplus _{j=1}^m R/(s-\lambda _j)^{a_j},\) 
where \(\lambda _j \)  are non-zero complex numbers and \(a_j\)  are positive integers. It turns out that the set of eigenvalues of \(h^i\)  can be recovered as
\((A):=\lbrace  \lambda _j \rbrace _{1\le j \le m},\) 
and the maximal size of the Jordan blocks is
\(J(A):= \max _{1 \le j \le m} \lbrace a_j\rbrace .\) 
Now we list two simple facts related to finite generated torsion \(R\) -modules, where the proofs are left to the reader. 
Let \(A \overset{u}{\rightarrow } B \overset{v}{\rightarrow } C\)  be a complex of finitely generated torsion \(R\) -modules. Then \(v/ u\)  is also a finitely generated torsion \(R\) -module and we have that
(a)
\((v/ u) \subset ( B) \) ,
(b)
\(J(v/ u) \le J(B) \) .

Let \(0 \rightarrow A \rightarrow B \rightarrow C\rightarrow 0\)  be a short exact sequence of finitely generated torsion \(R\) -modules. Then we have that
(a)
\((B) = ( A) \cup (C) \) ,
(b)
\(\max \lbrace  J(A), J(C)\rbrace  \le J(B) \le J(A)+J(C) \) .

The explanation in the beginning of this section shows that the spectral sequence
\( E_1^{i,j}= \mathcal {H}^i\left(\phi _\pi \left(\varepsilon ^{}_* {D^{j+1}(F)}\right)\right)_0 \Rightarrow \tilde{H}^{i+j}(Y_\delta ).\) 
can be studied in the category of \(R\) -modules, where every entry in it is a torsion \(R\) -module.
The assumption that \(f\)  has corank one implies that the one-parameter unfolding \(F\)  of \(f\)  also has corank one. Since \(F\)  is stable by assumption, the spaces \(D^{k}(F)\)  are smooth and have the right dimension. The generic fibre \(D^k(f_\delta )\)  coincides with the Milnor fiber of the germ of maps:
\(D^{k+1}(F) \rightarrow T.\) 
Then the classical monodromy theorem about eigenvalues and size of Jordan blocks apply to the monodromy automorphisms \(H^i( D^{k} (f_\delta ))\rightarrow H^i( D^{k} (f_\delta ))\) . Hence, by Lemma , the size of the Jordan blocks of the monodromy automorphism \(H^i_{}( D^{k} (f_\delta ))\rightarrow H^i_{}( D^{k} (f_\delta ))\)  is at most \(i+1\) , i.e.,
\(J(E^{i,k-1}_1)\le i+1\) 
Since the page \(E^{\bullet , \bullet }_{r+1}\)  is computed from \(E^{\bullet , \bullet }_{r}\)  by taking cohomologies, we conclude from Lemma  that the infinity page satisfies the above property as well.
Note that \(E_1^{i,k-1}=0\)  when \(k\le 1\) .
By the isomorphism \(H^\ell (Y_\delta )\cong \bigoplus _i E^{i,\ell -i}_\infty \)  and Lemma , we obtain that
\(J(H^\ell (Y_\delta ))\le \sum _{0\le i <\ell } J(E^{i,\ell -i}_\infty ) \le \sum _{0\le i < \ell } (i+1)= \ell (\ell +1)/2 .\) 
When \(f\)  has only isolated instability locus and \(p>n+1\) , then Proposition  gives us that \(E_1^{i,k-1}\)  are concentrated in degrees \( i=kn-(k-1)p\) . In particular, \(p>n+1\)  implies that the spectral sequence degenerates at \(E_1\) -page. Hence the
possibly non-trivial reduced cohomologies are
\(\tilde{H}^{kn-(k-1)(p-1)}(Y_\delta )\cong \tilde{H}^{kn-(k-1)p}_{}(D^k(f_\delta ))\)  for \(1< k \le \left\lfloor \frac{p}{p-n} \right\rfloor \)
with the corresponding size of Jordan block at most \(kn-(k-1)p+1\) .
Observe that the proof of Theorem  can be used to show
that the eigenvalues of \(h^i\colon H^i (Y_\delta )\rightarrow H^i (Y_\delta )\)  are
roots of unity, directly from the classical result about eigenvalues
of the monodromy of hypersurfaces (that is, with no resort to the more
general result of Lê for fibrations on analytic spaces). Moreover,
it shows that the eigenvalues of the monodromy on \(H^i (Y_\delta )\)  are
a subset of the union of eigenvalues of the monodromies on \(H^i_{}(D^k(f_\delta )),k\ge 0\) .
We saw earlier in Remark REF 
how the study of Milnor fibers of hypersurfaces
\(g \colon ({n+1},0) \rightarrow (0)\)  was equivalent to the
study of disentanglements of bi-germs \(f \colon (n,\lbrace p,q\rbrace )\rightarrow ({n+1},0)\) .
The above considerations show that this also comprises the monodromy:
Since all the ingredients involved in the spectral sequence admit an \(R\) -module structure,
the isomorphism \(\tilde{H}^i(M)\cong \tilde{H}^{i+1}(Y_\delta )\)  is an isomorphism of
\(R\) -modules and therefore the monodromies on both sides are compatible
by the considerations in Remark REF .
